In addition, when such constraints are satisfied, then each marginal component of the MAL in (REF ) follows a univariate AL distribution, that is \(Y_{tj} \sim \textnormal {AL}(\mu _{tj}, \tau _j, \delta _{tj})\) , where \(\delta _{tj}\)  represents the time-varying scale parameter of \(Y_{tj}\) . This allows us to exploit the result of [1]}, who shows the link between the scale parameter of the AL distribution and the ES risk measure in a univariate framework. By extending these results, we provide new insights on how to estimate conditional VaR and ES jointly in a multidimensional setting, blackthat accounts for correlations between marginals. This is explained in details in the next section.
